and
\(\lbrace x_j^\tau \rbrace _{j=1}^{d+1}\)  are the vertices of the \(d\) -simplex \(\tau \in \mathcal {T}_h\) , and \(\mathcal {T}_h\)  is a regular partition of \(\Omega \) . Hence, our numerical technique for tackling the difficult \(q_h\) -subproblem via solving (REF ) can be represented as replacing the problem (REF ) with (REF ), in which the objective function is regularized by a semi-proximal regularization term. In other words, applying the active-set Newton method along with the Schur complement reduction can be explained as replacing the optimization problem (REF ) by the proximally regularized one (REF ). Replacing the \(q_h\) -subproblem (REF ) with (REF ) in the ADMM (REF ) hence results in the so-called proximal ADMM, which has been well studied in the optimization area. We refer to, e.g., [1]}, [2]}, for convergence of various proximal versions of the ADMM.
